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International Journal of Mechanical and Materials Engineering (IJMME), Vol.5 (2010), No.2, 163-170.
REYNOLDS AND PRANDTL NUMBERS EFFECTS ON MHD MIXED CONVECTION IN A LID-DRIVEN CAVITY ALONG WITH JOULE HEATING AND A CENTERED HEAT CONDUCTING CIRCULAR BLOCK
M. M. Rahman1, M. M. Billah2, M. A. H. Mamun 3, R. Saidur4 and M. Hasanuzzaman4 1Department of Mathematics 1Bangladesh University of Engineering and Technology (BUET), Dhaka-1000, Bangladesh 2Department of Arts and Sciences, Ahsanullah University of Science and Technology (AUST), Dhaka, Bangladesh 3The Institute of Ocean Energy, Saga University, Japan 4Department of Mechanical Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia Email: [email protected]
Received 20 May 2010, Accepted 29 July 2010
ABSTRACT Nu average Nusselt number p dimensional pressure (Nm-2) The fluid flows and heat transfer induced by the combined P dimensionless pressure effects of mechanically driven lid and buoyancy force Pr Prandtl number within a rectangular cavity is investigated in this paper Ra Rayleigh number numerically. The horizontal walls of the enclosure are Re Reynolds number insulted while the right vertical wall is maintained at a Ri Richardson number uniform temperature higher than the left vertical wall. In T dimensional temperature(K) addition, it contains a heat conducting horizontal circular ∆T temperature difference (K) block in its centre. The governing equations for the problem u, v dimensional velocity components (ms-1) are first transformed into a non-dimensional form and the U, V dimensionless velocity components resulting nonlinear system of partial differential equations U0 lid velocity are solved by using the finite element formulation based on V cavity volume (m3) the Galerkin method of weighted residuals. The analysis is x, y Cartesian coordinates (m) conducted by observing the variations of streamlines and X, Y dimensionless cartesian coordinates isotherms for different Reynolds number and Prandtl number ranging from 50 to 200 and from 0.071 to 3.0 Greek symbols respectively. The results indicated that both the streamlines α thermal diffusivity (m2s-1) and isotherms strongly depend on the Reynolds number and β thermal expansion coefficient (K-1) Prandtl number. Moreover, the results of this investigation ν kinematic viscosity (m2s-1) are also illustrated by the variations of average Nusselt θ dimensionless temperature number on the heated surface and average fluid temperature μ dynamic viscosity (m2s-1) in the enclosure. ρ density of the fluid (kgm-3) -1 -1 Keywords: Mixed convection, Galerkin method, Circular σ fluid electrical conductivity (Ω .m ) block and Enclosure. ψ stream function Subscripts NOMENCLUTURE av average 2 B0 magnetic induction (Wb/m ) h heated wall Cp Specific heat of fluid at constant pressure c cold wall D block of diameter s solid -2 g gravitational acceleration (ms ) Gr Grashof number h convective heat transfer coefficient(Wm–2K–1) 1. INTRODUCTION Ha Hartmann number 1.1 Literature Review J Joule heating parameter -1 -1 kf thermal conductivity of the fluid (Wm K ) Flow and heat transfer phenomena in lid-driven enclosures -1 -1 ks thermal conductivity of the solid (Wm K ) caused by the conjugate effect of buoyancy and shear forces. K solid fluid thermal conductivity ratio Mixed convection flow and heat transfer in lid-driven L length of the cavity (m) enclosures have been receiving a considerable attention in the literature. Conjugate mixed convection heat transfer has 163 possible applications in many engineering and natural configuration the isothermal sidewalls of the cavity were processes. Such applications include cooling of electronic moving downwards with uniform velocity while the top wall devices, lubrication technologies, drying technologies, food was adiabatic. Oztop and Dagtekin (2004) investigated processing, and flow and heat transfer in solar ponds, numerically steady state two-dimensional mixed convection thermal –hydraulics of nuclear reactors etc. Flow and heat problem in a vertical two-sided lid-driven differentially transfer from obstructed enclosures are often encountered in heated square cavity. many engineering applications to enhance heat transfer such A combined free and forced convection flow of an as micro-electronic device, flat plate solar collectors, and electrically conducting fluid in cavities in the presence of a flat-plate condensers in refrigerator etc. This kind of magnetic field is of special technical significance because of problems was investigated mostly for the case of natural its frequent occurrence in many industrial applications such convection in enclosures. Roychowdhury et al. (2002) as geothermal reservoirs, cooling of nuclear reactors, analyzed the natural convective flow and heat transfer thermal insulations and petroleum reservoirs. These types of features for a heated cylinder kept in a square enclosure with problems also arise in electronic packages, microelectronic different thermal boundary conditions. Dong and Li (2004) devices during their operations. Garandet et al. (1992) and Hasanuzzaman et al. (2007) studied conjugate of natural studied natural convection heat transfer in a rectangular convection and conduction in a complicated enclosure. The enclosure with a transverse magnetic field. Rudraiah et al. authors investigated the influences of material character, (1995a) investigated the effect of surface tension on geometrical shape and Rayleigh number on the heat transfer buoyancy driven flow of an electrically conducting fluid in a in overall concerned region and concluded that the flow and rectangular cavity in the presence of a vertical transverse heat transfer increase with the increase of thermal magnetic field to see how this force damps hydrodynamic conductivity in the solid region; both geometric shape and movements. At the same time, Rudraiah et al. (1995b) also Rayleigh number affect the overall flow and heat transfer studied the effect of a magnetic field on free convection in a significantly. Hasanuzzaman et al. (2009) investigated the rectangular enclosure. The problem of unsteady laminar nature convection in closed cavity of refrigerator. Braga and combined forced and free convection flow and heat transfer Lemos (2005) numerically studied steady laminar natural of an electrically conducting and heat generating or convection within a square cavity filled with a fixed amount absorbing fluid in a vertical lid-driven cavity in the presence of conducting solid material consisting of either circular or of a magnetic field was formulated by Chamkha (2002). square obstacles. The authors showed that the average Recently, Rahman et al. (2009a) studied MHD mixed Nusselt number for cylindrical rods is slightly lower than convection around a heat conducting horizontal circular those for square rods. Lee and Ha (2005) investigated cylinder placed at the center of a rectangular cavity along natural convection in a horizontal layer of fluid with a with joule heating. Very recent, the effect of a heat conducting body in the interior, using an accurate and conducting horizontal circular cylinder on MHD mixed efficient Chebyshev spectral collocation approach. Later on, convection in a lid-driven cavity along with joule heating is the same authors Lee and Ha (2006) also studied natural investigated by Rahman et al. (2009b). The authors convection in horizontal layer of fluid with heat generating concluded the streamlines and isotherms strongly depend on conducting body in the interior. Kumar and Dalal (2006) the size and locations of the inner cylinder, but the thermal studied natural convection around a tilled heated square conductivity of the cylinder has significant effect only on cylinder kept in an enclosure in the range of 103 ≤ Ra ≤ 106. the isothermal lines. The authors reported detailed flow and heat transfer features for two different thermal boundary conditions and found 1.2 Objectives of the Present Study that the uniform wall temperature heating is quantitatively In this paper, the effect of Reynolds and Prandtl number on different from the uniform wall heat flux heating. MHD mixed convection in a lid-driven rectangular cavity The cavity with moving lid has the most important along with a heat conducting circular block is to analyze application for these heat transfer mechanism, which is seen using Finite Element technique. From the literature review it in cooling of electronic chips, solar energy collection and is clearly seen that no work has been paid on such research food industry etc. Numerical analysis of these kinds of interest. systems can be found in many literatures. Moallemi and Jang (1992) investigated mixed convective flow in a bottom 2. PHYSICAL DOMAIN heated square lid-driven cavity. The authors studied the The configuration under study with the system of co- effect of Prandtl number on the flow and heat transfer ordinate is sketched in Figure 1. Fluid flows and heat process. They found that the effects of buoyancy are more transfer modeled in a two-dimensional rectangular lid- pronounced for higher values of Prandtl number and also driven cavity of length L with a heat conducting horizontal derived a correlation for the average Nusselt number in circular block of diameter D = 0.2, placed at the centre of terms of the Prandtl number, Reynolds number and the cavity. The left wall of the cavity is set to move upward Richardson number. Aydin and Yang (2000) numerically on its own plane at a velocity U0. Horizontal walls of the studied mixed convection heat transfer in a two-dimensional cavity are insulated while the left and right vertical walls are square cavity having an aspect ratio of 1. In their isothermal but the temperature of the right wall is higher 164 than that of the left wall. The fluid permeated by a uniform UUPUU 1 22 external magnetic field B . The resulting convective flow is UV (2) 0 22 governed by the combined mechanism of driven (share and X Y X Re XY buoyancy) force and the electromagnetic retarding force. V V P1 2 V 2 V Ha 2 The magnetic Reynolds number is assumed to be small so U V Ri V (3) 22 that the induced magnetic field produced by the motion of X Y Y ReXY Re the electrically conducting fluid is negligible compared to the applied magnetic field B0. The density variation The thermal energy transport equation considered only in the body force term according to the 1 22 Boussinesq approximation. In addition, joule heating is UVJV 2 (4) considered, but pressure work, radiation and viscous 22 X Y Re Pr XY dissipation are supposed to be negligible. All solid boundaries are assumed to be rigid no-slip walls. For the solid circular block 22 ss U0 0 (5) XY22 The following non-dimensional variables are used in the g adiabatic above equations. y Th x y u v p XYUVP,,,,, x LLUU 2 00U0 L B 0 TTTTc s c Tc , TTTTs h c h c The dimensionless parameters that have been appeared in the above equations are:
L 32 2 Re U0 L , Gr g TL , Ri Gr Re , Figure 1 Schematic representation of the physical model 2 2 2 2 Ha B0 L , Pr , J B00 LU Cp T , with boundary conditions K ksf k 2.1 Mathematical Modeling Here Re , Gr, Ri, Ha and Pr are the familiar Reynolds, The flow of enclosed fluid is considered to be tow- Grashof, Richardson, Hartmann and Prandtl numbers dimensional, steady, incompressible and laminar. In respectively. We propose that J be called the Joule heating addition, the electrically conducting fluids are assumed to be parameter. We also propose that K be called solid fluid Newtonian with all the fluid properties taken as except for thermal conductivity ratio. the density variation with temperature for Boussinesq 2.2 Boundary Conditions approximation. The governing equations for the system of enclosed fluid are the expression for the conservation of The dimensionless form of the boundary conditions, mass, momentum and energy transport at every point of the associated to the problem are: system within the limit of basic assumptions and the use of UV0, 1, 0 on the left vertical wall appropriate boundary conditions. Moreover, the equation of continuity and u-momentum equation remain unchanged, UV0, 0, 1 on the right vertical wall but the equation of v-momentum is modified from Maxwell’s field equation and Ohm’s law. On the other hand, UV0, 0 on the solid surface the equation of energy is modified due to joule heating considered in the cavity. UV0, 0, 0 at the top and bottom walls N The governing equations are non-dimensionalzed to yield s The continuity equation K at the fluid-solid interface NNfluid solid UV 0 (1) XY Where N is the non-dimensional distances either along X or Y direction acting normal to the surface. The momentum equations
165 The average Nusselt number at the heated wall of the cavity 3.1 Effect of Reynolds number based on the dimensionless quantities may be defined by The effect of Reynolds number on the flow fields as 1 streamlines in the cavity operating at three different values Nu dY and the average temperature of the fluid in X of Ri, while the values of Pr = 0.71 is keeping fixed 0 presented in the Figure 2. As well known from the literature, the cavity is defined by av d V/ V , where V is the the values of the Richardson number Ri is a measure of the importance of natural convection to forced convection. Left cavity volume. column of this Figure shows that the forced convection The dimensionless stream function is defined as plays a dominant role; as a result the recirculation flow is mostly generated only by the moving lids at low Ri (Ri = UV, . 0.0) and all the values of Re considered, as expected which YX is due to the upward motion of the left wall. It is also observed that the orientation of the core in the recirculation 2.3 Numerical Procedure cell changes as Reynolds number Re changes. Further at Ri The governing equations together with the boundary = 1.0, two counter rotating cells are developed in the cavity conditions are solved numerically by a Galarkin finite for all the values of Re, which indicates both the buoyant element method. The method for analyzing mixed force and the shear driven force are presented in the cavity. convection in an obstructed vented cavity in our recent In this folder at low Re = 50 the clockwise cell, which is due previous study Rahman et al. (2009c) is employed in order to shear driven force occupies most of the part of the cavity to investigate the mixed convection in an obstructed lid- and two small anticlockwise rotating cells due to buoyant driven cavity with slight modification. force are developed at the bottom and top corner in the cavity near the right wall. But, dramatically, the In order to ensure the grid-independence solutions, a series anticlockwise rotating cell is reduced with increasing Re. of calculation were conducted for different grid Moreover, Bouncy dominated streamlines are observed at Ri distributions. The details of grid independence test are = 5.0 for different values of Re. available in Rahman et al. (2009b). The corresponding effect of the Reynolds number on The accuracy of the numerical model was also checked by thermal fields as isotherms at various values of Ri is shown comparing the results from the present study with those in the Figure 3. From this figure, we can ascertain that for Ri obtained by Chamkha (2002). The comparison is well listed = 0.0 and Re = 50, the isothermal lines near the hot wall are in Rahman et al. (2009a). parallel to the heated surface and parabolic shape isotherms 3. RESULTS AND DISSCUSSION are seen at the left top corner in the cavity, which is similar to forced convection and conduction like distribution. It is The main objective of this investigation is to analyze the also seen that isothermal lines start to turn back from the effects of Re and Pr on the flow and heat transfer in a lid- cold wall to the hot wall near the top wall at Ri = 0.0 and all driven cavity. The mixed convection phenomenon inside an values of Re due to the dominating influence of conduction obstructed lid-driven cavity is influenced by the Reynolds and forced convection in the upper part of the cavity. number Re, Prandtl number Pr, Richardson number Ri, Further at Ri = 1.0, both the buoyant force and the shear Hartmann number Ha, Joule heating parameter J, solid fluid force are same order of magnitude, the isothermal lines are thermal conductivity ratio K. In order to focus on the effect nearly parallel to the vertical walls in the cavity and of Re and Pr at the three convective regimes in the cavity, parabolic shape isotherms at the left top corner in the cavity we assign Ha =10.0, J =1.0, K =5.0. In addition, the effect becomes negligible for the lower values of Re (Re = 50, of cavity aspect ratio and Hartmann number on the flow and 100) and parabolic shape isotherms are seen at the right top heat transfer characteristics have already been reported by corner in the cavity for the higher values of Re (Re = 150, Rahman et al. (2009a). Very Recent, the effect of the size, 200). Moreover, the convective distortion of isothermal lines location and the thermal conductivity of the inner cylinder start to appear at Re = 50 and Ri = 5.0. Next at Ri = 5.0 and on the flow and heat transfer in the cavity have been Re = 100, it is seen that the isothermal lines turn back recorded by Rahman et al. (2009b). The results are towards the left cold wall near the top of the cavity and a presented in terms of streamline and isotherm patterns at the thermal boundary layer is developed near the left vertical three different regimes of flow, viz., pure forced convection, (cold) wall due to the dominating influence of the mixed convection and dominating natural convection with convective current in the upper part of the cavity. In Ri = 0.0, 1.0 and 5.0 respectively. The variations of the addition, at Ri = 5.0, the convective distortion in the average Nusselt number at the heated surface and average isotherms become more and the thermal boundary layer near fluid temperature in the cavity are plotted for the different the cold wall becomes more concentrated with further values of the parameters. In addition, the variations of the increasing the values of Reynolds number due to the strong average Nusselt number at the heated surface are also influence of the convective current. decorated in tabular form.
166
= = 200
= = 200
Re Re
= = 150
= = 150
Re
Re
= = 100
= = 100
Re Re
= = 50
= = 50
Re Re
Ri = 0.0 Ri =1.0 Ri = 5.0 Ri = 0.0 Ri =1.0 Ri = 5.0
Figure 2 Streamlines for different values of Re and Ri, Figure 3 Isotherms for different values of Re and Ri, while Pr = 0.71. while Pr = 0.71
The effect of Reynolds number on average Nusselt number Nu at the heated surface and average fluid temperature in the cavity are displayed as a function av of Richardson number at some particular Reynolds number in the Figure 4. It is observed that the average Nusselt number at the hot wall decreases very sharply in
the forced convection dominated region and increases gradually in the free convection dominated region with increasing Ri for the higher values of Reynolds number
Re (Re = 100, 150 and 200), but is different for the lowest value of Reynolds number Re (Re = 50). However, maximum values of Nu is found for the highest value of Re (Re = 200) at the aforesaid three convective
regimes. In addition, the average fluid temperature av in the cavity increases smoothly for higher values of Re (Re
= 100, 150, 200) and increases slowly for the lowest value of Re (Re = 50) with increasing Ri. It is also added that the values of are found minimum for Re = 200 at av the pure forced convection (Ri = 0.0) and for Re = 50 at the pure mixed convection (Ri = 1.0) and free convection dominated region. Finally, the quantitative differences of the values of Nu at different values of Re are documented Figure 4 Effect of Reynolds number Re on (i) average in Table 1. Nusselt number and (ii) average fluid temperature while Pr = 0.71.
167 Table 1 Variation of average Nusselt number with cavity. It is also seen that the shear effect produces the Reynolds number. clockwise vortex, which is comparatively smaller than that of the two-cellular counter clockwise vortex produced by
Nu the buoyant force. As the Richardson number increases Ri further to 5.0, the heat transfer is mostly by convection in Re = 50 Re = 100 Re = 150 Re = 200 the cavity, as a result the two-cellular counter clockwise 0.0 1.259431 1.710726 2.113681 2.440418 vortex become uni-cellular and large enough and the 1.0 1.086457 1.022651 1.120741 1.252180 clockwise vortex becomes shrink in size at each values of Pr considered. Besides, the flows represented by the 2.0 0.992551 1.064352 1.220219 1.351091 streamlines are almost independent of the Prandtl number at 3.0 0.973097 1.132416 1.303509 1.451683 each Ri. 4.0 0.982533 1.182924 1.367189 1.526451 5.0 0.997365 1.217358 1.410237 1.572461
= = 3.0 3.2 Effect of Prandtl number
The effect of Prandtl number on the flow fields as Pr
streamlines in the cavity at three different values of Ri is shown in figure 5, while Re = 100.
= = 1.0
Pr
= = 3.0
Pr
= 0.71
Pr = = 1.0
Pr
= 0.071
Pr
= = 0.71
Pr Ri = 5.0 Ri = 0.0 Ri =1.0
Figure 6 Isotherms for different values of Pr and Ri, while Re = 100.
= = 0.071
The effect of Prandtl number on thermal characteristics as Pr isotherms in the cavity at three different values of Ri are shown in figure 6, while Re = 100. The isotherms at very
Ri = 0.0 Ri =1.0 Ri = 5.0 low Pr (Pr = 0.071) and lower Ri (Ri = 0.0, 1.0) become almost parallel to the vertical walls, resembling the Figure 5 Streamlines for different values of Pr and Ri, conduction like heat transfer in the cavity. A closer while Re = 100 examinations also show the isothermal lines are symmetric about the line Y = 0.5. As Ri increases to 5.0, the symmetry in isotherms become disappears in the cavity. But in this The flow fields for all values of Pr (Pr = 0.071, 0.71, 1.0 folder the degree of distortion from the conduction heat and 3.0) at low Ri (= 0.0) are found to be established due to transfer is very noticeable and the isotherms become more the shear induced force by the moving lid only. Next at Ri = packed near the left top surface in the cavity at each values 1.0, the balance between the shear and buoyancy effect is of Ri and the higher Prandtl numbers (Pr = 0.71, 1.0 and manifested in the formation of two vortices inside the
168 3.0). The bend in isothermal lines appears due to the high Table 2 Variation of average Nusselt number with convective current inside the cavity. Figures 7 depict the Prandtl number. variations of average Nusselt number Nu at the heated wall, average temperature av of the fluid in the cavity at various Nu values of Pr and Ri. It is shown that, for the lowest Pr the Ri average Nusselt number decreases gradually with increasing Pr = 0.071 Pr = 0.71 Pr = 1.0 Pr = 3.0 Ri and for the higher Pr (Pr = 0.71 and 1.0) the values of Nu decreases with increasing Ri in the forced convection 0.0 1.039148 1.710726 2.041835 3.136956 dominated region and increases gradually with Ri in the free 1.0 1.024186 1.022651 1.071852 1.422283 convection dominated region. But for the highest Pr (Pr = 3.0), it is seen that the values of Nu decreases sharply with 2.0 0.986682 1.064352 1.136079 1.623106 increasing Ri in the forced convection dominated region and increases smoothly up to Ri 1.8, after then Nu is 3.0 0.929720 1.132416 1.203577 1.616093 independent of Ri. In addition, maximum values of Nu are found for the highest value of Pr at all values of Ri. 4.0 0.857280 1.182924 1.262789 1.619037 Moreover, the average fluid temperature av in the cavity 5.0 0.773427 1.217358 1.311545 1.642432 increase smoothly for higher values of Pr (Pr = 0.71, 1.0 and 3.0) and increases gradually for the lowest value of Pr (Pr = 0.071) with increasing Ri. On the other hand, 4. CONCLUSION minimum values of av are found at the highest Pr (Pr = 3.0) in the forced convection dominated region and at the lowest The present study investigates numerically the value of Pr (Pr = 0.071) in the free convection dominated characteristics of a two dimensional MHD mixed- region. Lastly, the quantitative differences of the values of convection problem in a lid-driven square cavity with a heat Nu at different values of Pr are tabulated in Table 2. conducting horizontal solid circular block. To simulate the flow and heat transfer in the cavity, Finite Element Method is implemented here. A detailed analysis for the distribution of streamlines, isotherms, average Nusselt number at the heated surface and average temperature of the fluid are carried out to investigate the effect of the Reynolds number and Prandtl number on the fluid flow and heat transfer in the
cavity for different Richardson numbers in the range of 0.0 Ri 5.0. The following conclusions are made: Reynolds number Re has a great significant effect on the streamlines and isotherms at the three convective regimes. Buoyancy-induced vortex in the streamlines increased and thermal layer near the cold surface become thin and concentrated with increasing Re. Reynolds number Re has also a great significant effect on the average Nusselt
numbers, Nu and the average temperature, av in the cavity.
The influence of Prandtl number on the streamlines in the cavity is found insignificant for all the values of Ri, whereas the influence of Pr on the isotherms is remarkable for different values of Ri. In addition, for lower values of Pr the heat transfer is dominated by conduction and it become reduces with increasing Pr. The average Nusselt number is
always superior for the larger values of Pr. The average temperature of the fluid in the cavity is inferior for Pr = 3.0 in the forced convection dominated region, also for Pr
=0.071 in the free convection dominated region.
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